ó î&]\c@`s¿dZddlmZmZmZddlZddlmZm Z m Z m Z ddl m Z mZddd d gZdddd „Zd „Zd „Zd„Zd efd„ƒYZdS(s2Nearly exact trust-region optimization subproblem.i(tdivisiontprint_functiontabsolute_importN(tnormtget_lapack_funcstsolve_triangulart cho_solvei(t_minimize_trust_regiontBaseQuadraticSubproblemt_minimize_trustregion_exactt estimate_smallest_singular_valuetsingular_leading_submatrixtIterativeSubproblemc K`s^|dkrtdƒ‚n|dkr6tdƒ‚nt||d|d|d|dt|S(s Minimization of scalar function of one or more variables using a nearly exact trust-region algorithm. Options ------- initial_tr_radius : float Initial trust-region radius. max_tr_radius : float Maximum value of the trust-region radius. No steps that are longer than this value will be proposed. eta : float Trust region related acceptance stringency for proposed steps. gtol : float Gradient norm must be less than ``gtol`` before successful termination. s9Jacobian is required for trust region exact minimization.s?Hessian matrix is required for trust region exact minimization.targstjacthesst subproblemN(tNonet ValueErrorRR (tfuntx0R RRttrust_region_options((s@lib/python2.7/site-packages/scipy/optimize/_trustregion_exact.pyR s  cC`s°tj|ƒ}|j\}}||kr9tdƒ‚ntj|ƒ}tj|ƒ}xt|ƒD]}d|||j||f}d|||j||f}||d|j|dd…|f|}||d|j|dd…|f|} t|ƒt |dƒt|ƒt | dƒkrO|||<|||d)qd|||<| ||d)qdWt ||ƒ} t | ƒ} t |ƒ} | | } | | }| |fS(sYGiven upper triangular matrix ``U`` estimate the smallest singular value and the correspondent right singular vector in O(n**2) operations. Parameters ---------- U : ndarray Square upper triangular matrix. Returns ------- s_min : float Estimated smallest singular value of the provided matrix. z_min : ndarray Estimatied right singular vector. Notes ----- The procedure is based on [1]_ and is done in two steps. First it finds a vector ``e`` with components selected from {+1, -1} such that the solution ``w`` from the system ``U.T w = e`` is as large as possible. Next it estimate ``U v = w``. The smallest singular value is close to ``norm(w)/norm(v)`` and the right singular vector is close to ``v/norm(v)``. The estimation will be better more ill-conditioned is the matrix. References ---------- .. [1] Cline, A. K., Moler, C. B., Stewart, G. W., Wilkinson, J. H. An estimate for the condition number of a matrix. 1979. SIAM Journal on Numerical Analysis, 16(2), 368-375. s.A square triangular matrix should be provided.iiÿÿÿÿN( tnpt atleast_2dtshapeRtzerostemptytrangetTtabsRR(tUtmtntptwtktwptwmtpptpmtvtv_normtw_normts_mintz_min((s@lib/python2.7/site-packages/scipy/optimize/_trustregion_exact.pyR .s," --2      cC`sttj|ƒ}tj|ƒ}tjtj|ƒddƒ}tj|||ƒ}tj|||ƒ}||fS(s Given a square matrix ``H`` compute upper and lower bounds for its eigenvalues (Gregoshgorin Bounds). Defined ref. [1]. References ---------- .. [1] Conn, A. R., Gould, N. I., & Toint, P. L. Trust region methods. 2000. Siam. pp. 19. taxisi(RtdiagRtsumtmintmax(tHtH_diagt H_diag_abst H_row_sumstlbtub((s@lib/python2.7/site-packages/scipy/optimize/_trustregion_exact.pytgershgorin_bounds}s cC`sÑtj|d|d…|dfdƒ||d|df}t|ƒ}tj|ƒ}d||d<|dkrÇt|d|d…d|d…f|d|d…|df ƒ||d*n||fS(s  Compute term that makes the leading ``k`` by ``k`` submatrix from ``A`` singular. Parameters ---------- A : ndarray Symmetric matrix that is not positive definite. U : ndarray Upper triangular matrix resulting of an incomplete Cholesky decomposition of matrix ``A``. k : int Positive integer such that the leading k by k submatrix from `A` is the first non-positive definite leading submatrix. Returns ------- delta : float Amount that should be added to the element (k, k) of the leading k by k submatrix of ``A`` to make it singular. v : ndarray A vector such that ``v.T B v = 0``. Where B is the matrix A after ``delta`` is added to its element (k, k). Nii(RR/tlenRR(tARR#tdeltaR R(((s@lib/python2.7/site-packages/scipy/optimize/_trustregion_exact.pyR ’sA  QcB`sJeZdZdZejeƒjZdddd„Z d„Z d„Z RS(sÔQuadratic subproblem solved by nearly exact iterative method. Notes ----- This subproblem solver was based on [1]_, [2]_ and [3]_, which implement similar algorithms. The algorithm is basically that of [1]_ but ideas from [2]_ and [3]_ were also used. References ---------- .. [1] A.R. Conn, N.I. Gould, and P.L. Toint, "Trust region methods", Siam, pp. 169-200, 2000. .. [2] J. Nocedal and S. Wright, "Numerical optimization", Springer Science & Business Media. pp. 83-91, 2006. .. [3] J.J. More and D.C. Sorensen, "Computing a trust region step", SIAM Journal on Scientific and Statistical Computing, vol. 4(3), pp. 553-572, 1983. g{®Gáz„?gš™™™™™¹?gš™™™™™É?cC`sßtt|ƒj||||ƒd|_d|_d|_||_||_t d|j fƒ\|_ t |j ƒ|_ t|j ƒ\|_|_t|j tjƒ|_t|j dƒ|_|j |j|j|_dS(Niÿÿÿÿitpotrftfro(R<(tsuperR t__init__tprevious_tr_radiusRt lambda_lbtnitertk_easytk_hardRRtcholeskyR9t dimensionR8thess_gershgorin_lbthess_gershgorin_ubRRtInfthess_infthess_frotEPSt CLOSE_TO_ZERO(tselftxRRRthesspRCRD((s@lib/python2.7/site-packages/scipy/optimize/_trustregion_exact.pyR?×s     cC`sãtd|j|t|j |j|jƒƒ}tdt|jjƒƒ |j|t|j|j|jƒƒ}||j kr–t|j |ƒ}n|dkr«d}n+tt j ||ƒ||j ||ƒ}|||fS(séGiven a trust radius, return a good initial guess for the damping factor, the lower bound and the upper bound. The values were chosen accordingly to the guidelines on section 7.3.8 (p. 192) from [1]_. i(R1tjac_magR0RGRKRJRtdiagonalRHR@RARtsqrtt UPDATE_COEFF(RNt tr_radiust lambda_ubRAtlambda_initial((s@lib/python2.7/site-packages/scipy/optimize/_trustregion_exact.pyt_initial_valuess  cC`sT|j|ƒ\}}}|j}t}t}d|_xötr.|rNt}nA|j|tj|ƒ}|j|dtdtdtƒ\} } |jd7_| dkræ|j |j kræt | tf|j ƒ} t | ƒ} | |kr|dkrt}Pnt| | ddƒ} t | ƒ}| |d| ||}||}| |kr°t| ƒ\}}|j| ||ƒ\}}t||gd tƒ}tj| tj|| ƒƒ}|d|d|||d}||jkrú| ||7} Pn|}t|||dƒ}|j|tj|ƒ}|j|dtdtdtƒ\}} | dkrs|}t}qãt||ƒ}ttj||ƒ||j||ƒ}q+t| |ƒ|}||jkr×Pn|}|}q9| dkrÁ|j |j krÁ|dkr)tj|ƒ} t}Pnt| ƒ\}}|}|d|d|j||dkrv||} Pn|}t|||dƒ}ttj||ƒ||j||ƒ}q9t|| | ƒ\}}t |ƒ}t||||dƒ}ttj||ƒ||j||ƒ}q9W||_||_||_| |fS( sSolve quadratic subproblemitlowert overwrite_atcleanittransRitkey(RXRFtTruetFalseRBRRteyeRERQRMRRRRR tget_boundaries_intersectionsR0RtdotRDR1RSRTRCRR RAtlambda_currentR@(RNRURcRARVR t hits_boundarytalready_factorizedR2RtinfoR!tp_normR"R*t delta_lambdat lambda_newR+R,ttattbtstep_lentquadratic_termtrelative_errortcR;R(R)((s@lib/python2.7/site-packages/scipy/optimize/_trustregion_exact.pytsolves’         "    '     N( t__name__t __module__t__doc__RTRtfinfotfloattepsRLRR?RXRp(((s@lib/python2.7/site-packages/scipy/optimize/_trustregion_exact.pyR ¼s( ((Rst __future__RRRtnumpyRt scipy.linalgRRRRt _trustregionRRt__all__RR R R8R R (((s@lib/python2.7/site-packages/scipy/optimize/_trustregion_exact.pyts "  O  *